Interfacial Tension in Binary Polymer Blends and the Effects of Copolymers as Emulsifying Agents - Polymer Thermodynamics - page 196

interphase. At and near the plane AA 0 , the properties of S are identical with those of
the bulk phase . However, moving from AA 0 to BB 0 within the region S represents

a gradual change in the properties of the interphase, from those corresponding to
phase

a

.
In the bulk phases, the force across any unit area is equal in all directions, as is
the hydrostatic pressure P . In the interphase, the force is not the same in all
directions. However, if a plane of unit area is chosen parallel to AA 0 or BB 0 , the
force across the plane is the same for any position of the plane whether it lies in

a to those corresponding to phase b

,
or S, because hydrostatic changes are assumed negligible. In contrast, the force
balance for planes that cross the interphase, i.e., perpendicular to AA 0 , is altered by
the inclusion of an additional term due to the interfacial tension, g . This force is
associated with the anisotropic nature of intermolecular forces that result from the
concentration gradient within the interphase.
The influence of the interfacial tension term on the thermodynamics can be
illustrated by considering the work, W , performed on the interphase when addi-
tional interphase is formed. If the interphase volume increases by d V S , i.e., a
thickness increase of d x and an area increase of d A S , the force balance leads to:

a , b

PA S dx dA S

W ¼ ð Px gÞ (10a)

or:

PdV S þ g dA S W ¼

(10b)

This last expression is the analogous work term for an interphase, which
corresponds to the three-dimensional P d V term for a bulk phase. Incorporation

of this term into the first and second laws of thermodynamics for multiconstituent
open systems results in:

X

PdV S þ g dA S dU ¼ TdS þ m i dn i (11)

i

where T is the thermodynamic temperature, S is the entropy, U is the internal
energy, and m i and n i are the chemical potential and number of moles of type i.
Integration of the above equation, at constant intensive variables, produces the
corresponding Euler relationship:

!

X

PV S A S g ¼ U þ TS m i dn i = (12)

i

Therefore, g is the excess free energy per unit area arising from the formation of
the interphase; it is equal to the difference between the Gibbs free energy of the
system with the interphase, ( U + PV S TS ), and that of an identical system without